Abstract
In this paper, we are concerned with the fractional order equations (1) with Hartree type \begin{document}$ \dot{H}^{\frac{α}{2}} $\end{document} -critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). Then, by applying the method of moving planes in integral forms, we prove that positive solutions \begin{document}$ u $\end{document} to (1) and (3) are radially symmetric about some point \begin{document}$ x_{0}∈\mathbb{R}^{d} $\end{document} and derive the explicit forms for \begin{document}$ u $\end{document} (Theorem 1.3 and Corollary 1). As a consequence, we also derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2).
Highlights
In this paper, we consider the following-critical fractional order equation with Hartree type nonlinearity (−∆) α 2 u = 1 |x|2α ∗|u|2 u, x ∈ Rd, u ∈ H α 2(Rd), u(x) > 0, x ∈ Rd, where
Applying the regularity lifting lemma by contracting operators, we prove the following regularity lifting theorem for (3) which indicates that weak solutions are smooth
U is radially symmetric and monotone decreasing about some point x0 ∈ Rd, in particular, the positive solution u must assume the following form d−α u(x) = μ 2 Q μ(x − x0)
Summary
-critical nonlinearity and its equivalent integral equations (3). We first prove a regularity result which indicates that weak solutions are smooth (Theorem 1.2). By applying the method of moving planes in integral forms, we prove that positive solutions u to (1) and (3) are radially symmetric about some point x0 ∈ Rd and derive the explicit forms for u (Theorem 1.3 and Corollary 1). We derive the best constants and extremal functions in the corresponding Hardy-Littlewood-Sobolev inequalities (Corollary 2)
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